## Sample assignment on R statistics help

Answer all questions. Marks are indicated beside each question. You should submit your solutions before the

You should submit both

• a .pdf file containing written answers (word processed, or hand-written and scanned), and

• an .R file containing R code.

For all answers include

• the code you have written to determine the answer, the relevant output from this code, and a justification of how you got your answer.

• Total marks: 60 1. Consider the one parameter family of probability density functions

`fb for − b ≤ x ≤ b`

where b > 0.

(a) Write R code to plot this pdf for various values of b > 0. [2 marks]

(b) Determine the method of moments estimator for the parameter b. (No R code necessary) [4 marks]

(c) Determine the Likelihood function for the parameter b. By writing R code to plot a suitable graph, determine that the derivative of this likelihood function is never zero. [4 marks] (d) Hence find the Maximum Likelihood Estimator for the parameter b. (No R code necessary) [4 marks] (e) The data in the file Question 1 data.csv contains 100 independent draws from the probability distribution with pdf fb(x), where the parameter b is unknown. Load the data into R using the command

D <- read . csv (path_to_f i l e )$x

where path_to_file indicates the path where you have saved the .csv file

e.g. path_to_file = “c:/My R Downloads/Question 1 data.csv”

Note that forward slashes are used to indicate folders (this is not consistent with the usual syntax for Microsoft operating systems).

Write R code to calculate an appropriate Method of Moments Estimate and a Maximum Likelihood Estimate for the parameter b, given this data. [4 marks]

- The data in the file Question 2 data.csv is thought to be a realisation of Geometric Brownian Motion

St = S0eσWt+µt

where Wt is a Wiener process and σ,µ and S0 are unknown parameters. Load the data into R using the command

S <- read . csv (path_to_f i l e )

where path_to_file indicates the path where you have saved the .csv file.

(a) Write R code to determine the parameter S0. [2 marks]

(b) Write R code to determine if Geometric Brownian Motion is suitable to model this data.

You may do this by

• plotting an appropriate scatter plot/histogram, and/or • using an appropriate statistical test.

[6 marks]

(c) Write R code to determine an estimate for µ and σ2 using Maximum Likelihood Estimators.

(You do not have to derive these estimators). [5 marks] - The data in the file Question 3 data.csv is a matrix of transition probabilities of a Markov Chain. Load the data into R using the command

P <- as . matrix ( read . csv (path_to_f i l e ))

with an appropriate value for path_to_file.

(a) Verify that this Markov Chain is ergodic. (No R code necessary) [4 marks]

(b) Suppose that an initial state vector is given by

x=(0.1,0.2,0.4,0.1,0.2) (1)

Write R code to determine the state vector after 10 time steps. Do this without diagonalising the matrix

P. [3 marks]

(c) Write R code to verify this answer by diagonalising the matrix P. Note that the eigen(A) function produces the right-eigenvectors of a matrix A (solutions of Av = λv) However we want the left-eigenvectors (solutions of vA= λv).

These are related by

v is a left-eigenvector of A if and only if vT is a right-eigenvector of AT.

[8 marks]

(d) Hence, or otherwise, determine the limiting distribution with the initial state vector given in (1).

[4 marks] - The data if the file Question 4 data.csv is a generator matrix for a Markov Process. Load the data into R using the command

A <- as . matrix ( read . csv (path_to_f i l e ))

with an appropriate value for path_to_file.

(a) Suppose that X0 =0. Write R code to simulate one realisation of the Markov Process Xt. The output should be two vectors (or one data frame with two variables).

• The first vector indicates transition times.

• The second vector indicates which state the Markov Process takes at this time (i.e. one of 0,1,2,3,4).

How to proceed:

• The first line of your code must read

set . seed (4311)

to ensure that this realisation is repeatable.

• For each Xt you must determine

– what the transition time s to the next state is,

– what the probabilities to transfer to each state are, and hence randomly select a suitable value for Xt+s.

[9 marks] (b) Write R code to plot an appropriate graph that describes this realisation. [1 mark]